"Is there a topological groupoid structure on the pair $(Gl(n,\mathbb{R}), O(n))$, with their standard topologies?"
This is already asked here but this linked question is a very general question, so we consider its special case about general linear group, as an independent question.
There is an easy differentiable groupoid structure: Use the Iwasawa decomposition $G=KAN= K\times A\times N$. In the case of the question this is the Gram-Schmidt orthogonalization proceedure for the rows and remembering the coefficients as an upper triangular matrix with positive entries on the main diagonal: $GL(n)\ni g = k.a.n\in O(n)\times \mathbb R_{>0}^n\times N$ where $N$ are upper triangular matrices with 1 on the main diagonal. We take $k.a.n \mapsto k$ as source and target map, and the groupoid structure is then $(k.a_1.n_1,k.a_2.n_2)\mapsto k.a_1.n_1.a_2.n_2$.
